Problem 25 - Calculus for Biology and Medicine

Question

To determine the germination success of seeds of a certain plant, you plant 162 seeds. You find that 117 of the seeds germinate. Estimate the probability of germination and give a \(95 \%\) confidence interval.

Step-by-Step Solution

Answer

The estimated probability of germination is about 0.7222 with a 95% confidence interval of [0.6523, 0.7921].

Step 1: Determine the Probability of Germination

The probability of a seed germinating is calculated by dividing the number of seeds that germinated by the total number of seeds planted. So, the probability \( p \) is given by \( p = \frac{117}{162} \). Calculating this gives \( p \approx 0.7222 \).

Step 2: Identify the Formula for Confidence Interval

The formula for a confidence interval for a proportion \( p \) is given by \[ \hat{p} \pm Z \cdot \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}, \] where \( \hat{p} \) is the estimated probability of germination, \( Z \) is the z-value corresponding to the desired confidence level, and \( n \) is the total number of trials.

Step 3: Calculate the Z-value for 95% Confidence

For a 95% confidence interval, the z-value is approximately 1.96 because the standard normal distribution table lists a z-value of 1.96 for 95% confidence.

Step 4: Calculate the Standard Error

The standard error \( SE \) is part of the confidence interval equation and is computed using: \[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}. \] Substitute \( \hat{p} = 0.7222 \) and \( n = 162 \): \[ SE = \sqrt{\frac{0.7222(1 - 0.7222)}{162}} \approx \sqrt{\frac{0.7222 \times 0.2778}{162}} \approx 0.0357. \]

Step 5: Calculate the Confidence Interval

Using the values from the previous steps, the confidence interval is \[ 0.7222 \pm 1.96 \times 0.0357. \] This is \[ 0.7222 \pm 0.0699. \] Therefore, the 95% confidence interval is approximately \( [0.6523, 0.7921] \).

Key Concepts

Probability of Germination

The probability of germination is a way to express how likely it is for a seed to sprout into a plant. To find this probability, we use the ratio of the number of seeds that actually germinated to the total number of seeds planted. In our exercise, 117 out of 162 seeds germinated.
Therefore, we calculate the probability of germination as follows:

• Take the number of germinated seeds (117).
• Divide by the total number of seeds (162).

This gives a probability: \[ p = \frac{117}{162} \approx 0.7222 \]This means there is approximately a 72.22% chance for any given seed to germinate.
Understanding this probability helps us evaluate the expected success of planting more seeds.

Z-value

The Z-value is a critical number in statistics used to determine the confidence level of an estimate. When calculating confidence intervals, it acts as a multiplier that amplifies the standard error. This gives us a range within which we expect the true proportion to lie.
For a 95% confidence level, which is standard in many applications, the Z-value is approximately 1.96. This value comes from standard normal distribution tables, which reflect the probability of observing a value within this range from a normal distribution.
Using a Z-value of 1.96 implies we are 95% confident that the true proportion of seed germination falls within our calculated interval. This confidence is vital for making predictions and making informed agricultural decisions.

Standard Error

The standard error (SE) is a statistical measure that helps us understand how much variability there is in our sample proportion. Simply put, it tells us how well the sample proportion (\( \hat{p} \) ) estimates the true population proportion. The formula for standard error with proportions is:\[ SE = \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]Where:- \( \hat{p} \) is the sample proportion (probability of germination in our case).- \( n \) is the total number of seeds planted.
Plugging in the numbers:

• Our estimated probability of germination, \( \hat{p} \approx 0.7222 \), and
• Total seeds \( n = 162 \).

This gives us:\[ SE \approx \sqrt{\frac{0.7222 \cdot (1 - 0.7222)}{162}} \approx 0.0357 \]The standard error of 0.0357 reflects the sampling variability, showing how our single sample proportion likely deviates from the true underlying population proportion.

Proportion Estimation

Proportion estimation involves determining the likelihood or ratio of a specific outcome in a sample, which in this case is the germination of seeds. We begin with calculating the sample proportion (\( \hat{p} \) ) by dividing the number of germinated seeds by the total number planted. Once we have \( \hat{p} \) , the next step is to determine how confident we are that \( \hat{p} \) accurately estimates the true probability of germination in the population of all similar seeds. To do this, we calculate a confidence interval. This interval provides a range of values within which the true proportion is likely to lie. The formula used is:\[ \hat{p} \pm Z \cdot SE \] - The \( Z \) -value helps adjust our confidence level (95% in this exercise).- \( SE \) is the standard error.
Using our example:

• \( \hat{p} = 0.7222 \) ,
• \( Z = 1.96 \) , and
• \( SE = 0.0357 \).

The computed confidence interval is:\[ [0.7222 - 1.96 \times 0.0357, 0.7222 + 1.96 \times 0.0357] \approx [0.6523, 0.7921] \]This means we are 95% confident the true proportion of seeds that will germinate falls between 65.23% and 79.21%, a useful insight for planning future planting.